Could we extend isomorphisms between cohomologies of h-injective complexes to h-injective complexes themselves?

Let $$R$$ be an associative ring with unit and $$I$$ be a complex of $$R$$-modules. We call $$I$$ is h-injective if for any acyclic complex $$T$$ of $$R$$-modules, the mapping complex $$\text{Hom}_R(T,I)$$ is acyclic too.

Now let $$I$$ and $$J$$ two h-injective complexes of $$R$$-modules. Let $$H^{\cdot}(I)$$ and $$H^{\cdot}(J)$$ be the cohomologies of $$I$$ and $$J$$.

My question is: suppose we have maps of $$R$$-modules $$f^n:H^n(I)\overset{\sim}{\to}H^n(J)$$ for each $$n$$, could we define a quasi-isomorphism $$\phi: I\overset{\sim}{\to} J$$ such that $$H^n(\phi)=f^n$$ for each $$n$$? We may add some condition such as both $$I$$ and $$J$$ are componentwisely injective and have bounded cohomologies.